Polytope of Type {168}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {168}*336
Also Known As : 168-gon, {168}. if this polytope has another name.
Group : SmallGroup(336,93)
Rank : 2
Schlafli Type : {168}
Number of vertices, edges, etc : 168, 168
Order of s0s1 : 168
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{168,2} of size 672
{168,4} of size 1344
{168,4} of size 1344
{168,4} of size 1344
{168,4} of size 1344
Vertex Figure Of :
{2,168} of size 672
{4,168} of size 1344
{4,168} of size 1344
{4,168} of size 1344
{4,168} of size 1344
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {84}*168
3-fold quotients : {56}*112
4-fold quotients : {42}*84
6-fold quotients : {28}*56
7-fold quotients : {24}*48
8-fold quotients : {21}*42
12-fold quotients : {14}*28
14-fold quotients : {12}*24
21-fold quotients : {8}*16
24-fold quotients : {7}*14
28-fold quotients : {6}*12
42-fold quotients : {4}*8
56-fold quotients : {3}*6
84-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {336}*672
3-fold covers : {504}*1008
4-fold covers : {672}*1344
5-fold covers : {840}*1680
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 6)( 4, 5)( 8, 15)( 9, 21)( 10, 20)( 11, 19)( 12, 18)
( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)
( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)
( 63, 72)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)
( 92,141)( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)( 99,134)
(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,148)(107,154)
(108,153)(109,152)(110,151)(111,150)(112,149)(113,162)(114,168)(115,167)
(116,166)(117,165)(118,164)(119,163)(120,155)(121,161)(122,160)(123,159)
(124,158)(125,157)(126,156);;
s1 := ( 1, 93)( 2, 92)( 3, 98)( 4, 97)( 5, 96)( 6, 95)( 7, 94)( 8, 86)
( 9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)( 16, 99)
( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)( 24,119)
( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)( 32,111)
( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)( 56,150)
( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,135)
( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)
( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)( 80,147)
( 81,146)( 82,145)( 83,144)( 84,143);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(168)!( 2, 7)( 3, 6)( 4, 5)( 8, 15)( 9, 21)( 10, 20)( 11, 19)
( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)
( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)
( 54, 81)( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)
( 62, 73)( 63, 72)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)
( 91,128)( 92,141)( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)
( 99,134)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,148)
(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,162)(114,168)
(115,167)(116,166)(117,165)(118,164)(119,163)(120,155)(121,161)(122,160)
(123,159)(124,158)(125,157)(126,156);
s1 := Sym(168)!( 1, 93)( 2, 92)( 3, 98)( 4, 97)( 5, 96)( 6, 95)( 7, 94)
( 8, 86)( 9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)
( 16, 99)( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)
( 24,119)( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)
( 32,111)( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)
( 56,150)( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)
( 64,135)( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)
( 72,127)( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)
( 80,147)( 81,146)( 82,145)( 83,144)( 84,143);
poly := sub<Sym(168)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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